EA hasn't released any games "with a single use DLC code that's required to input before starting a new game" that requires you to be logged into PSN or XBL in order to start a new game.
They're certainly crippling the single player experiences in their games (Such as the limited selection of golf courses in their most recent golf game, with most being DLC add-ons), but they haven't done that. Their games can still be played on offline condoles and you can still start new games while offline.